Contests for catch shares

Abstract

We study contests in which fishers (or players), in a fishery managed under a catch-share program, compete over catch shares by expending irreversible effort, and the fishery manager, influenced by the players’ effort, allocates their catch shares. We first show that the number of active players and their identities in each period depend only on the players’ marginal costs, and they remain exactly the same across all periods. Then, we show that, in each period, an active player with a lower marginal cost expends greater effort and secures a greater catch share than a player with a higher marginal cost. We also show that a player who expends zero effort due to his relatively high marginal cost ends up with a less and less catch share over time.

Appendices

Appendix A: The sign of $\partial \pi ^*/\partial \theta$

Let $Z=\sum \nolimits _{k \in N^*} (G_k^2/H_k)-\sum \nolimits _{j \in N}(p-c_j)/n$. Since the term $\{ 1+2\delta (1-\theta )\}$ in (12) is always positive, the sign of $\partial \pi ^* / \partial \theta$ is the same as the sign of Z.

Consider the case where $c_1 < c_j =c$ for $j=2, \ldots , n$. In this case, we have that $n^*=n$. It is straightforward to obtain

$$\begin{aligned} Z&=\sum \limits _{j \in N} (G_j^2/H_j)- \sum \limits _{j \in N} (p-c_j)/n \\&=[(p-c_1)\{(n-1)(p-c_1)-(n-2)(p-c)\}^2+(n-1)(p-c)^3]/\{ (p-c) \\&\quad +\, (n-1)(p-c_1)\}^2 -\{(p-c_1)+(n-1)(p-c)\}/n. \end{aligned}$$

(A1)

Let $B=p-c$. Then we have that $p-c_1= \lambda (p-c)=\lambda B$, where $\lambda >1$. We can then rewrite (A1) as

$$\begin{aligned} Z=[\lambda B \{(n-1)\lambda -(n-2)\}^2+(n-1)B]/\{1+(n-1)\lambda \}^2-\{\lambda B +(n-1)B\}/n. \end{aligned}$$

It is straightforward to see that the sign of Z is the same as the sign of $Z_1$:

$$\begin{aligned} Z_1&=(n-1)^2\lambda ^3-3(n-1)^2\lambda ^2+(n^2-5n+3)\lambda +n-1 \\&=(n-1)^2\lambda ^2(\lambda -3)+(n^2-5n+3)\lambda +n-1. \end{aligned}$$

(A2)

Appendix B: Analysis of the game with multiple periods

In this appendix, we analyze the game with multiple periods presented in Sect. 4. To obtain outcomes associated with an SPNE of the game, we work backward. We first analyze the proper subgames starting at the beginning of period T, then look at the strategic interaction among the players in period $T-1$, then in period $T-2$, and so on.

Let $x_{it}^e$, for $i \in N$ and for $t=1, \ldots , T$, denote player i’s effort level in period t in the equilibrium of the $(s_{1t-1}, \ldots , s_{nt-1})$ subgame starting at the beginning of period t. Let $\mathbf {x_t^e} \equiv (x_{1t}^e, \ldots , x_{nt}^e)$ for $t=1, \ldots , T$.

In period T, knowing his fractional catch share granted in period $T-1$, player i, for $i \in N$, seeks to maximize his payoff $\pi _i^T$ over his effort level $x_{iT}$, given his belief about the other players’ effort levels in period T:

$$\begin{aligned} \pi _i^T =\pi _{iT}= (p-c_i) \{\theta f_i(\mathbf {x_T})+(1-\theta )s_{iT-1} \}Q -x_{iT}. \end{aligned}$$

In period $T-1$, knowing his fractional catch share in period $T-2$, player i, for $i \in N$, seeks to maximize his payoff $\pi _i^{T-1}$ over his effort level $x_{iT-1}$, given his belief about the other players’ effort levels in period $T-1$:

$$\begin{aligned} \pi _i^{T-1}&= (p-c_i) s_{iT-1} Q -x_{iT-1} + \delta [(p-c_i) \{ \theta f_i(\mathbf {x_T^e})+(1-\theta )s_{iT-1}\}Q-x_{iT}^e]\\&=(p-c_i) \{ 1+ \delta (1-\theta )\} \{ \theta f_i (\mathbf {x_{T-1}})+(1-\theta )s_{iT-2}\}Q -x_{iT-1}+R_i^{T-1}, \end{aligned}$$

where $R_i^{T-1} = \delta \{ (p-c_i)\theta f_i(\mathbf {x_T^e})Q -x_{iT}^e \}$. It is straightforward to see that $R_i^{T-1}$ does not depend on $x_{iT-1}$, and is nonnegative.

In period $T-2$, knowing his fractional catch share granted in period $T-3$, player i, for $i\in N$, seeks to maximize his payoff $\pi _i^{T-2}$ over his effort level $x_{iT-2}$, given his belief about the other players’ effort levels in period $T-2$:

$$\begin{aligned} \pi _i^{T-2}&= (p-c_i)s_{iT-2} Q -x_{iT-2} +\delta [(p-c_i) \{ \theta f_i(\mathbf {x_{T-1}^e})+(1-\theta )s_{iT-2}\}Q-x_{iT-1}^e] \\&\quad +\, \delta ^2 [(p-c_i) \{ \theta f_i(\mathbf {x_T^e})+(1-\theta )s_{iT-1}\}Q-x_{iT}^e] \\&= (p-c_i) \{ 1+ \delta (1-\theta )+\delta ^2(1-\theta )^2\}\{ \theta f_i(\mathbf {x_{T-2}})\\ {}&\quad +(1-\theta )s_{iT-3}\}Q-x_{iT-2} + R_i^{T-2}, \end{aligned}$$

where $R_i^{T-2}=\delta \{ (p-c_i)\theta f_i(\mathbf {x_{T-1}^e})Q-x_{iT-1}^e\} + \delta ^2(p-c_i)(1-\theta )\theta f_i(\mathbf {x_{T-1}^e})Q + \delta ^2 \{ (p-c_i)\theta f_i(\mathbf {x_T^e})Q-x_{iT}^e\}$. It is straightforward to see that $R_i^{T-2}$ does not depend on $x_{iT-2}$, and is nonnegative.

Similarly, in period t, for $t=2, \ldots , T-3$, knowing his fractional catch share granted in period $t-1$, player i, for $i \in N$, seeks to maximize his payoff $\pi _i^t$ over his effort level $x_{it}$, given his belief about the other players’ effort levels in period t:

$$\begin{aligned} \pi _i^t=(p-c_i) \left\{ \sum \limits _{z=t}^T \delta ^{z-t}(1-\theta )^{z-t}\right\} \{ \theta f_i(\mathbf {x_t})+(1-\theta )s_{it-1}\}Q-x_{it} + R_i^t, \end{aligned}$$

where $R_i^t$ is the sum of the terms which do not depend on $x_{it}$. It is straightforward to see that $R_i^t$ is nonnegative.

In period 1, knowing his initial (fractional) catch share, player i, for $i \in N$, seeks to maximize his overall payoff $\pi _i^1$ over his effort level $x_{i1}$, given his belief about the other players’ effort levels in period 1:

$$\begin{aligned} \pi _i^1=(p-c_i) \{ \sum \limits _{z=t}^T \delta ^{z-1}(1-\theta )^{z-1}\}\{ \theta f_i(\mathbf {x_1})+(1-\theta )(1/n)\}Q-x_{i1} + R_i^1, \end{aligned}$$

where $R_i^1$ is the sum of the terms which do not depend on $x_{i1}$. It is straightforward to see that $R_i^1$ is nonnegative.

Note that $\pi _i^t$, for $t=1, \ldots , T$, is strictly concave in $x_{it}$, which implies that the second-order condition for maximizing $\pi _i^t$ is satisfied. Let $X_t^e = \sum \limits _{j \in N} x_{jt}^e$. Player i’s effort level, $x_{it}^e$, satisfies either (B1) or (B2):

$$\begin{aligned} (p-c_i) \{ \sum \limits _{z=t}^T \delta ^{z-t}(1-\theta )^{z-t}\}\theta Q \{ (X_t^e-x_{it}^e)/(X_t^e)^2\}-1=0 \text { for } x_{it}^e >0 \end{aligned}$$

(B1)

or

$$\begin{aligned} (p-c_i) \{ \sum \limits _{z=t}^T \delta ^{z-t}(1-\theta )^{z-t}\}\theta Q \{ (X_t^e-x_{it}^e)/(X_t^e)^2\}-1\le 0 \text { for } x_{it}^e = 0. \end{aligned}$$

(B2)

Let $n_t^e$, for $t=1, \ldots , T$, represent the number of the active players in period t in the equilibrium of the $(s_{it-1}, \ldots , s_{nt-1})$ subgame starting at the beginning of period t. Let $N_t^e \equiv \{ 1, \ldots , n_t^e\}$. Using (B1) and (B2), we obtain Lemma B1.

Lemma B1

• (a) In period t, for $t=1, \ldots , T$, active player k chooses

  $$\begin{aligned} x_{kt}^e= & {} \theta \left\{ \sum \limits _{z=t}^T \delta ^{z-t}(1-\theta )^{z-t} \right\} (n_t^e-1) [(p-c_k) \sum \limits _{j \in N_t^e} \{ 1/(p-c_j)\}\\&\quad -\, n_t^e +1]Q/(p-c_k)\left[ \sum \limits _{j \in N_t^e}\{ 1/(p-c_j)\}\right] ^2, \end{aligned}$$

  for $k \in N_t^e$, where $(p-c_{n_t^e}) \sum \nolimits _{j \in N_t^e} \{ 1/(p-c_j) \}+n_t^e+1>0$.

• (b) In period t, player h, if any, chooses $x_{ht}^e=0$, for $h \in N {\setminus } N_t^e$, in which case

  $$\begin{aligned} (p-c_h)\sum \limits _{j \in N_t^e} \{ 1/ (p-c_j)\} -n_t^e +1 \le 0. \end{aligned}$$

• (c) The total effort level of the players in period t is

  $$\begin{aligned} X_t^e = \theta \left\{ \sum \limits _{z=t}^T \delta ^{z-t}(1-\theta )^{z-t}\right\} (n_t^e-1)Q/\sum \limits _{j \in N_t^e} \{ 1/(p-c_j)\}. \end{aligned}$$

Lemma B1 shows that the effort level $x_{kt}^e$ of active player k in period t does not depend on the players’ (fractional) catch shares in period $t-1$. This implies that it does not depend on the players’ effort levels in period $t-1$. Lemma B1 also shows that the number of active players and their identities in period t depend only on the players’ marginal costs. Since each player has the same marginal cost in all T periods, and since an active player is determined by the same cutoff level of marginal cost in all T periods, it is straightforward to see that the number of active players and their identities remain exactly the same across all T periods.